The flavour of generalized cohomology which is both twisted as well as differential is called twisted differential cohomology (or: differential twisted cohomology).
A key motivation for twisted differential cohomology is its role in string theory as the type of generalized cohomology theory that classifies higher gauge fields (following Freed 00), notably in the example of twisted differential K-theory classifying the RR-field twisted by the B-field (see below).
Observing that
differential cohomology may be understood as being the intrinsic cohomology inside an (∞,1)-topos $\mathbf{H}$ which is cohesive (see at differential cohomology diagram);
twisted cohomology may be understood as being the intrinsic cohomology inside the tangent (∞,1)-topos $T \mathbf{H}$ of the given ambient (∞,1)-topos $\mathbf{H}$
and since
one may define twisted differential cohomology in generality to be the intrinsic cohomology inside cohesive tangent (∞,1)-toposes $T \mathbf{H}$ (Schreiber 13, Sec. 4.1.2, Schreiber 14, slide 16).
The objects of such a $T \mathbf{H}$ may be understood as sheaves of parametrized spectra (Braunack-Mayer 19, Theorem 3.4). For example with $\mathbf{H} =$ Smooth∞Grpd the objects of $T Smth\infty Grpd$ are smooth parametrized spectra. In this sense the formulation of twisted differential cohomology via cohesive tangent (∞,1)-topos generalizes the classical Brown representability theorem (identifying plain generalized (Eilenberg-Steenrod) cohomology theories with plain spectra) to twisted and differential cohomology.
By means of a suitable model category presentation of tangent (∞,1)-toposes, Braunack-Mayer 19, Example 3.23 proves that a more explicit component-based definition of twisted differential cohomology due to Bunke-Nikolaus 14 is a special case of this general definition from Schreiber 13, Sec. 4.1.2.
A simple example of twisted cohomology is twisted de Rham cohomology (see there for more), the twisted generalization of de Rham cohomology.
The archetypical example is twisted differential K-theory, which combines twisted K-theory with differential K-theory. To the extent that D-brane charge is classified by K-theory (see there), it is twisted differential K-theory that is relevant: the differential aspect captures the higher gauge field called the RR-field, and the twisted aspects captures the higher gauge field called the B-field, in string theory.
This example of D-brane charge used to be one of the main motivations for finding a definition and construction of twisted differential cohomology theories. Earlier account of D-brane charge theory had to assume without a construction that such a theory exists (e.g. DFM 09).
Other higher gauge fields in string theory/M-theory are expected to be cocycles in twisted differential cohomology theories for other generalized cohomology theories apart from K-thery. For instance the hypothesis that the M-theory C-field is topologically a cocycle in twisted cohomotopy (FSS19b, FSS19c) means that with all differential form data added. it is actually a cocycle in twisted differential cohomotopy.
The general abstract definition of twisted differential cohomology as the intrinsic cohomology of cohesive tangent (∞,1)-toposes is due to
Urs Schreiber, section 4.1.2 of Differential cohomology in a cohesive (∞,1)-topos, (arXiv:1310.7930)
Urs Schreiber, around slide 17 of Differential generalized cohomology in Cohesive homotopy type theory , talk at IHP trimester on Semantics of proofs and certified mathematics Workshop 1: Formalization of Mathematics, Institut Henri Poincaré, Paris, 5-9 May 2014 (pdf slides)
Daniel Grady, Hisham Sati, Twisted differential generalized cohomology theories and their Atiyah-Hirzebruch spectral sequence, Algebr. Geom. Topol. 19 (2019) 2899-2960 (arXiv:1711.06650, doi:10.2140/agt.2019.19.2899)
Domenico Fiorenza, Hisham Sati, Urs Schreiber, The character map in (twisted differential) non-abelian cohomology (arXiv:2009.11909)
Discussion of twisted differential K-theory along these lines:
Daniel Grady, Hisham Sati, Twisted differential KO-theory (arXiv:1905.09085)
(specifically: KO-theory)
A more component-based definition was given in
A proof that the definition of Bunke-Nikolaus 14 is a special case of that of Schreiber 13, Sec. 4.2.1 is due to
For more discussion of twisted differential cohomology as the intrinsic cohomology of Goodwillie-tangent spaces of cohesive (∞,1)-topos see there.